The story is the one about bad people doing bad things, and how they are responsible for the problems in the world. These bad people must be rooted out and stopped for the sake of the country and - often quite literally - the children.
A folklore term for this kind of story is a "subversion myth." Historical examples abound; here are a few from Jeffrey S. Victor's paper "Satanic Cult Rumors as Contemporary Legend," in Western Folklore 49:1 at page 51:
In Ancient Rome, the stories commonly claimed that Christians were kidnapping Roman children for use in secret ritual sacrifices. Later, during the Middle Ages, the stories claimed that Christian children were being kidnapped by Jews, again for use in secret ritual sacrifices....In France, just before the French Revolution, similar stories accused aristocrats of kidnapping the children of the poor, for use of their blood in medical baths. [Citations omitted.]
Police in modern Saudi Arabia routinely hunt, arrest, prosecute, and execute witches. In a similar vein, Christian religious programming on state-run television in Benin, Nigeria, in the early 1980s generally condemned Western influence and blamed the country's problems on corruption and witchcraft. "It is particularly appropriate that the church explains both individual and societal misfortune in simple, personalized terms, not only because such explanations are politically expedient and coincide with official doctrine, but also because they reflect the prevailing though patterns of those who still inhabit [a world of magic]," say Lyons & Lyons in "Magical Medicine on Television: Benin City, Nigeria," Journal of Ritual Studies 1:1 (1987).
The Benin example highlights that the vilified class need not exist in reality for the story to be effective, as with the satanic cult ritual abuse panics of the 1980s and to some extent the modern wars on terror and bigotry.
What makes the story of bad guys so popular? Mainly, it provides a universally applicable justification for why things are not going well. In human societies, things are generally not going well in a variety of ways, at least from the perspective of individual members, but subversion myths arise in times of special trouble; as Victor puts it, they "usually arise at times when a society is undergoing a deep cultural crisis of values, after a period of very rapid social change has caused much disorganization and widespread social stress."
This justification for things not going well satisfies people's need for an explanation for the troubles. A story that vilifies those in power may precede (and perhaps help bring about) the revolution, as in France. (Revolutions are particularly likely to occur when things are really, really not going well in a society in the first place, hence there is a great perceived need for an explanation.)
A story that vilifies others, however, is useful once the revolutionaries have seized power and become the government, with an interest in maintaining the status quo. When religious movements like Islamism, democracy, Christianity, and communism first come into power, things are generally pretty bad; the new government has something like regression to the mean on its side. Unfortunately, political societies are delicate, carefully evolved systems and it's amazing that they limp along at all; attempts at reform, like mutations in DNA, usually make things worse. The story offers hope. Millions of scapegoats have been executed by governments in service of this story.
Despite its flaws, the story is certainly more comfortable than suddenly accepting that large human societies just can't be very good or wise or fair or free, and that attempts to manipulate the intricate yet lumbering social ecosystem, no matter how well-meaning and carefully researched, usually make things worse.